If $R$ is a field, then one approach that explains at least some constructions is to extend a permutation module $P$ to a chain complex with semisimple homology, then resolve irreducible modules with permutation modules.  The second half, permutation resolutions of irreducible modules, then becomes a modified question that can be studied.

For instance in characteristic zero, many groups (or all of them?) have enough permutation modules to express every irrep as a formal linear combination of them; and any such formal linear combination can be dressed up to a resolution.  In characteristic 0, the question reduces to linear dependences among characters of permutation modules, which is a lame answer but at least valid for part of the question.

On that theme, [here is a preprint][1] by Boltje and Hartmann that constructs a conjectured resolution of Specht modules of $S_n$ (over $\mathbb{Z}$) by Young modules.  This is presumably a related tool.

  [1]: http://math.ucsc.edu/~boltje/publications/p09y.pdf